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A244979
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Decimal expansion of Pi/(2*sqrt(5)).
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4
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7, 0, 2, 4, 8, 1, 4, 7, 3, 1, 0, 4, 0, 7, 2, 6, 3, 9, 3, 1, 5, 6, 3, 7, 4, 6, 4, 3, 2, 0, 4, 8, 9, 4, 7, 9, 9, 4, 6, 6, 5, 0, 9, 1, 8, 7, 0, 6, 7, 2, 0, 2, 4, 1, 9, 9, 8, 9, 7, 2, 1, 0, 2, 6, 1, 9, 2, 1, 4, 1, 8, 8, 0, 6, 1, 9, 1, 8, 8, 2, 0, 5, 1, 0, 4, 1, 4, 2, 4, 1, 5, 3, 6, 5, 7, 6, 7, 2, 4, 0, 2, 1, 5, 0, 7
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OFFSET
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0,1
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REFERENCES
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George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), Chapter 13 A Master Formula, p. 250.
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LINKS
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FORMULA
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Equals Integral_(0..1) (1 + x^2)/(1 + 3*x^2 + x^4) dx.
Also equals beta(1/2, 1/2)/(2*sqrt(5)), where 'beta' is Euler's beta function.
Pi/(2*sqrt(5)) = Integral_{t = 0..a} (1 + t^2)*(1 + t^6)/(1 + t^10) dt = a + a^3/3 + a^7/7 + a^9/9 - a^11/11 - a^13/13 - a^17/17 - a^19/19 + ..., where a = 1/2(sqrt(5) - 1). Hint: differentiate atan( sqrt(5)*(t - t^3)/(1 - 3*t^2 + t^4) ). (End)
Equals (1/2)*Sum_{n >= 0} (-1)^n*( 1/(10*n + 1) + 1/(10*n + 3) + 1/(10*n + 7) + 1/(10*n + 9) ). Cf. A019692. - Peter Bala, Oct 30 2019
Equals Integral_{x=0..oo} 1/(x^2 + 5) dx.
Equals 0.1 * Integral_{x=0..oo} log(1 + 5/x^2) dx. (End)
Equals Integral_{x = 0..1} 2/(4*x^2 + 5*(1 - x)^2) dx. - Peter Bala, Jul 22 2022
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EXAMPLE
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0.702481473104072639315637464320489479946650918706720241998972102619214188...
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MATHEMATICA
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RealDigits[Pi/(2*Sqrt[5]), 10, 105] // First
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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